Speaking about ALL differential equations, it is extremely rare to find analytical solutions. Further, simple differential equations made of basic functions usually tend to have ludicrously complic...
Is there a reason it is so rare we can solve differential equations?
There are many different types or classifications of equations. Some examples include quadratic equations, polynomial equations, linear equations, parametric equations, differential equations, and many more.
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The right question is not "What is a differential?" but "How do differentials behave?". Let me explain this by way of an analogy. Suppose I teach you all the rules for adding and multiplying rational numbers. Then you ask me "But what are the rational numbers?" The answer is: They are anything that obeys those rules. Now in order for that to make sense, we have to know that there's at least ...
Then one thinks of differential operators as a linear maps between such spaces. Often the space of all linear maps between two spaces is itself a vector space and so one can indeed start to manipulate differential operators as if they are ‘objects’ in their own right eg add them together.